Related papers: Hypergeometric solutions to the symmetric q-Painle…

Polynomial invariants are fundamental objects in analysis on Lie groups and symmetric spaces. Invariant differential operators on symmetric spaces are described by Weyl group invariant polynomial. In this article we give a simple criterion…

The initial value spaces of the Painlev\'{e} equations are proposed by Okamoto. They are symplectic manifolds in which the Painlev\'{e} equations are described as polynomial Hamiltonian systems on all coordinates. In this article, we…

Hirota's discrete KdV equation is an integrable partial difference equation on $\mathbb{Z}^2$, which approaches the Korteweg-de Vries (KdV) equation in a continuum limit. In this paper, we show that its multiplicative-discrete versions have…

We give a new approach to the symmetries of the Painlev\'e equations $P_{V},P_{IV},P_{III}$ and $P_{II}$, respectively. Moreover, we make natural extensions to fourth-order analogues for each of the Painlev\'e equations $P_{V}$ and…

We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective…

A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear…

In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.

We obtain a comparison result for solutions to nonlinear fully anisotropic elliptic problems by means of anisotropic symmetrization. As consequence we deduce a priori estimates for norms of the relevant solutions.

Starting with a Riccati equation solved by hypergeometric functions, some sequences of rational solutions to Painleve' VI are obtained.

We find and study four kinds of a 4-parameter family of four-dimensional coupled Painlev\'e III systems with affine Weyl group symmetry of types $B_4^{(1)}$, $D_4^{(1)}$ and $D_5^{(2)}$. We also show that these systems are equivalent by an…

A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.

The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to…

We consider restriction analogues on hypersurfaces of the uniform Sobolev inequalities in Kenig, Ruiz, and Sogge and the resolvent estimates in Dos Santos Ferreira, Kenig, and Salo.

The question under consideration is Gevrey summability of power expansions of solutions to the third and fifth Painlev\'{e} equations near infinity. Methods of French and Japaneese schools are used to analyse these properties of formal…

We use the middle convolution to obtain some old and new algebraic solutions of the Painlev\'e VI equations.

The classical Painlev\'e equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the…

In this article, the structure of the Clifford-Weyl superalgebras and their associated Lie superalgebras will be investigated. These superalgebras have a natural supersymmetric inner product which is invariant under their Lie superalgebra…

We provide new insights into the solvability property of an Hamiltonian involving of the fourth Painlev\'e transcendent and its derivatives. This Hamiltonian is third order shape invariant and can also be interpreted within the context of…

Quasi-symmetric functions show up in an approach to solve the Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product…

We consider the problem of projecting a convex set onto a subspace, or equivalently formulated, the problem of computing a set obtained by applying a linear mapping to a convex feasible set. This includes the problem of approximating convex…